Higher accuracy analysis of instrumented indentation data obtained with pointed indenters

Abstract

We review current practice for describing force–displacement curves from pointed indenters, highlighting the consequences of the simplifications normally adopted. We derive two corrections, the ‘variable epsilon factor’ and the ‘radial displacement correction.’ These are especially important for highly elastic materials such as fused silica where the combined corrections can amount to 13% in the contact area, significantly increasing the accuracy of hardness and modulus results. In contrast, the so-called beta factor has minor importance. We compare our analytical results with finite element (FE) calculations and experimental results. Indenter area functions, obtained using the corrections, agree well with independent direct measurements by a traceably calibrated metrological atomic force microscope (AFM). Further formulae are derived to calculate the complete force–displacement curve of conical indenters and the indentation elastic and total energy. These formulae immediately identify a physical material limit above which a cone cannot generate plastic deformation; for a Berkovich indenter this is a hardness-to-modulus ratio of 0.18.